A minimum degree condition forcing a digraph to be k-linked

Given a digraph D, let δ0(D):=min{δ+(D),δ−(D)} be the minimum semi-degree of D. In [D. Kühn and D. Osthus, Linkedness and ordered cycles in digraphs, submitted] we showed that every sufficiently large digraph D with δ0(D)≥n/2+ℓ−1 is ℓ-linked. The bound on the minimum semi-degree is best possible and confirms a conjecture of Manoussakis [Y. Manoussakis, k-linked and k-cyclic digraphs, J. Combinatorial Theory B 48 (1990) 216-226]. We [D. Kühn and D. Osthus, Linkedness and ordered cycles in digraphs, submitted] also determined the smallest minimum semi-degree which ensures that a sufficiently large digraph D is k-ordered, i.e. that for every sequence s1,…,sk of distinct vertices of D there is a directed cycle which encounters s1,…,sk in this order.